Deflation Technique to Accelerate the Convergence of Iterative Solver for the Wave Scattering Problem A. H. Sheikh, A. G. Shaikh, Hisamuddin, Naeem Faraz, and Asif Ali Abstract—An iterative solution method for the discrete high wavenumber Helmholtz equation is presented. The basic idea for solution, already presented in [1], is to develop a precondi- tioner which is based on a Helmholtz operator with a complex- valued shift for a Krylov subspace iterative method. The preconditioner which can be seen as a strongly damped wave equation in Fourier space, can be approximately inverted by a multigrid method. Extensive deflation and spectral analysis, as Krylov subspace methods depends upon eigenvalues, highlights in this paper. Findings in analysis are validated by numerical results. Index Terms—Helmholtz equation, Multigrid Method, Pre- conditioning, Sparse linear systems, Deflation preconditioner. I. I NTRODUCTION W AVE scattering have many applications in physics, engineering and science. Examples include seismic imaging [1], [2], [3], [4], [5], radars, electromagnetism [6], bio medical imaging [7], (ultrasound), road-speed sensors etc. Wave scattering phenomena is mostly modeled by math- ematicians in the form of the Helmholtz equation [8], [9] and [10]. Solving Helmholtz equation requires the use of iterative methods. The Helmholtz equation in two dimensional (2D) or three dimensional (3D), the convergence is typically characterized by indefiniteness of the eigenvalues of the corresponding coefficient matrix. With such a property, an iterative method either basic or advanced, encounters conver- gence problems. The method usually converges very slowly or diverges [11]. There are very few choices of numerical methods to compute solution of very large sparse systems for many reasons, including memory, sparsity, heterogeneity of medium and indefiniteness. Indefiniteness limits the choice narrowly. The sparse direct solver have been used in [12], [13], [14] and [15]. They are heavily constrained with mem- ory and storage, hence are not practical for sufficiently large problems. The direct methods are not favorable for many obvious grounds, and they are too much time restrictions. They consume unaffordable memory for large problem which is under consideration. Discrete Helmholtz system, obtained by finite difference scheme, is approximated using Krylov subspace method. The preconditioner CSLP are tested with different shifts. Eigenvalue analysis of CSLP is given in Manuscript received July 03, 2019; revised August 08, 2019. A.H. Sheikh is with the Department of Mathematics & Statistics, QUEST Nawabshah, 67480 Pakistan e-mail: [email protected] . A. G. Shaikh is with department of Mathematics & Statistics, QUEST Nawabshah 67480, Pakistan e-mail: [email protected]. Hisamuddin is with department of Mathematics, Shah Abdul Latif Uni- versity Khairpur, 66020 Pakistan e-mail: [email protected]. Naeem Faraz is with Center of International Program, Donghua University Shanghai PRC e-mail: [email protected] . Asif Ali is with department of Basic Science & Related Studies, Mehran University of Engineering & Technology, Jamshoro 76062, Pakistan e-mail: [email protected] . accordance with solver performance. For small frequency, CSLP performs better whereas increasing frequency, CSLP becomes impractical in terms of memory and computational time. Deflation technique is used to address these types of issues. The Helmholtz equation can be read as -Δ 2 u(x, y) - k 2 (x, y)u(x, y)= f (x, y), (1) where Δ 2 = ∂ 2 (∂x 2 ) + ∂ 2 ∂y 2 , and u(x, y)the unknown variable, defined on the unit square domain Ω = (0, 1) × (0, 1), K wave number. The wave number k is related with wavelength λ as k(x, y)= 2π λ = omega c(x, y (2) where ω =2πF is angular velocity, F the wave frequency, λ = c(x,y) F the wavelength and c(x, y) is the speed of sound. A. Model Problem The Helmholtz problem considered this paper is non- homogeneous defined on the domain Ω=(x, y) × (x, y) where x, y ∈ (0, 1). The wavenumber is constant, indepen- dent of geometry. The source function is given as f (x, y)= δ(x, y)= δ(x - 1/2,y - 1/2). (3) With x, y ∈ (0, 1) where Dirac delta function is given as δ(x, y)= ( +∞ x =0,y =0 0 x 6=0,y 6=0 . (4) This source functions is used to model the source centered at ( 1 2 , 1 2 ). The domain is bounded by the Sommerfeld radiation conditions [6] [1] [16], which are given as ∂u ∂η - ιku =0. (5) This models the propagation of wave from center outwards direction.Discretization: two lines. The resultant linear sys- tem is written as A h u h = f h . (6) II. HELMHOLTZ SOLVERS Solving Helmholtz equation requires solution of resultant large sparse Linear System (6). For large, sparse matrix the Krylov subspaces are very popularchoice.The methods are developed on construction of iterants in the subspace.The space K j (A; r 0 )= Span{r 0 , Ar 0 ,A 2 r 0 , ··· ...A (j-1) r 0 }, is called the Krylov subspace of dimension j , associated with A and r 0 , and initial residual r 0 := g - Au 0 is related to the initial guess u 0 . Among methods which are based Proceedings of the World Congress on Engineering and Computer Science 2019 WCECS 2019, October 22-24, 2019, San Francisco, USA ISBN: 978-988-14048-7-9 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) WCECS 2019
Transcript
Deflation Technique to Accelerate the Convergenceof Iterative Solver for the Wave Scattering Problem
A. H. Sheikh, A. G. Shaikh, Hisamuddin, Naeem Faraz, and Asif Ali
Abstract—An iterative solution method for the discrete highwavenumber Helmholtz equation is presented. The basic ideafor solution, already presented in [1], is to develop a precondi-tioner which is based on a Helmholtz operator with a complex-valued shift for a Krylov subspace iterative method. Thepreconditioner which can be seen as a strongly damped waveequation in Fourier space, can be approximately inverted by amultigrid method. Extensive deflation and spectral analysis, asKrylov subspace methods depends upon eigenvalues, highlightsin this paper. Findings in analysis are validated by numericalresults.
Index Terms—Helmholtz equation, Multigrid Method, Pre-conditioning, Sparse linear systems, Deflation preconditioner.
I. INTRODUCTION
WAVE scattering have many applications in physics,engineering and science. Examples include seismic
imaging [1], [2], [3], [4], [5], radars, electromagnetism [6],bio medical imaging [7], (ultrasound), road-speed sensorsetc. Wave scattering phenomena is mostly modeled by math-ematicians in the form of the Helmholtz equation [8], [9] and[10]. Solving Helmholtz equation requires the use of iterativemethods. The Helmholtz equation in two dimensional (2D)or three dimensional (3D), the convergence is typicallycharacterized by indefiniteness of the eigenvalues of thecorresponding coefficient matrix. With such a property, aniterative method either basic or advanced, encounters conver-gence problems. The method usually converges very slowlyor diverges [11]. There are very few choices of numericalmethods to compute solution of very large sparse systemsfor many reasons, including memory, sparsity, heterogeneityof medium and indefiniteness. Indefiniteness limits the choicenarrowly. The sparse direct solver have been used in [12],[13], [14] and [15]. They are heavily constrained with mem-ory and storage, hence are not practical for sufficiently largeproblems. The direct methods are not favorable for manyobvious grounds, and they are too much time restrictions.They consume unaffordable memory for large problem whichis under consideration. Discrete Helmholtz system, obtainedby finite difference scheme, is approximated using Krylovsubspace method. The preconditioner CSLP are tested withdifferent shifts. Eigenvalue analysis of CSLP is given in
Manuscript received July 03, 2019; revised August 08, 2019.A.H. Sheikh is with the Department of Mathematics & Statistics, QUEST
Nawabshah, 67480 Pakistan e-mail: [email protected] .A. G. Shaikh is with department of Mathematics & Statistics, QUEST
Nawabshah 67480, Pakistan e-mail: [email protected] is with department of Mathematics, Shah Abdul Latif Uni-
versity Khairpur, 66020 Pakistan e-mail: [email protected] Faraz is with Center of International Program, Donghua University
Shanghai PRC e-mail: [email protected] .Asif Ali is with department of Basic Science & Related Studies, Mehran
University of Engineering & Technology, Jamshoro 76062, Pakistan e-mail:[email protected] .
accordance with solver performance. For small frequency,CSLP performs better whereas increasing frequency, CSLPbecomes impractical in terms of memory and computationaltime. Deflation technique is used to address these types ofissues.The Helmholtz equation can be read as
−∆2u(x, y)− k2(x, y)u(x, y) = f(x, y), (1)
where ∆2 = ∂2
(∂x2) + ∂2
∂y2 , and u(x, y)the unknown variable,defined on the unit square domain Ω = (0, 1) × (0, 1), Kwave number. The wave number k is related with wavelengthλ as
k(x, y) =2π
λ=omega
c(x, y(2)
where ω = 2πF is angular velocity, F the wave frequency,λ = c(x,y)
F the wavelength and c(x, y) is the speed of sound.
A. Model Problem
The Helmholtz problem considered this paper is non-homogeneous defined on the domain Ω = (x, y) × (x, y)where x, y ∈ (0, 1). The wavenumber is constant, indepen-dent of geometry. The source function is given as
f(x, y) = δ(x, y) = δ(x− 1/2, y − 1/2). (3)
With x, y ∈ (0, 1) where Dirac delta function is given as
δ(x, y) =
+∞ x = 0, y = 0
0 x 6= 0, y 6= 0. (4)
This source functions is used to model the source centered at( 12 ,
12 ). The domain is bounded by the Sommerfeld radiation
conditions [6] [1] [16], which are given as
∂u
∂η− ιku = 0. (5)
This models the propagation of wave from center outwardsdirection.Discretization: two lines. The resultant linear sys-tem is written as
Ahuh = fh. (6)
II. HELMHOLTZ SOLVERS
Solving Helmholtz equation requires solution of resultantlarge sparse Linear System (6). For large, sparse matrix theKrylov subspaces are very popularchoice.The methods aredeveloped on construction of iterants in the subspace.Thespace
is called the Krylov subspace of dimension j, associated withA and r0, and initial residual r0 := g − Au0 is relatedto the initial guess u0. Among methods which are based
Proceedings of the World Congress on Engineering and Computer Science 2019 WCECS 2019, October 22-24, 2019, San Francisco, USA
on construction the Krylov subspaces, a Conjugate Gradient(CG) [17],[18] [19] GMRES [20], CGS [21], Bi-CG [22],Bi-CGSTAB [23]and QMR are popular. For non-symmetriclinear systems, Krylov subspace can be built from Arnoldi’sprocess, which leads to GMRES(Saad and Schultz, 1986).GMRES is optimal method; it reduces the 2-norm of theresidual at the every iteration. GMRES, however, requirelong recurrences, which is usually limited by the availablememory. A remedy is by restarting, which some-times leadto slow convergence or stagnation [24].
A. Preconditioning
In an iterative method preconditioning is often vital com-ponent in enhancing the convergence of iterations, partic-ularly when the system is large sized.. There are manyiterative techniques for solving linear system. For large sparelinear system, the convergence rate is always a concern forresearchers; one can improve significantly the convergent rateby applying appropriate preconditioner. The convergence ofiterative methods depends on eigenvalues of the coefficientmatrix, it is often advantageous to use preconditioner thattransfer the system to one with a better distribution ofeigenvalues. The preconditioner is the key to successfuliterative solver. In brief, to make linear system favorable foriterative solver, the coefficient matrix is scaled with a matrixMcalled preconditioner. With choice of preconditionerMforLinear System 6, where the inverse of M is relativelyinexpensive to compute, and then the preconditioned systemis M ( − 1)Au = M ( − 1)f is supposed to be favorable foriterative solver. A few preconditioners have been tried forthe Helmholtz equation, for details see [10] [2] [4].
B. Complex Shifted Laplace Preconditioner
The Complex Shifted Laplace Preconditoner(CSLP) isthe discrete Helmholtz operator in addition with a complexshift (a, ιb). The CSLP is preconditioner based on operator,in contrast to decomposition type preconditioners, whichare matrix-based. The CSLP obtained by(finite difference)discretization of the shifted Helmholtz operator i.e.
M(a, b) := −∆− (a− ιb)k2, where a, b ∈ R,
where a and b are real and imaginary numbers respectively.The first precedent in operator based preconditioner for theHelmholtz equation was simple Laplace operator ∆, usedwithout any shifts. It works well, until mesh size is small.For large size of mesh, convergence starts to stagnates, andalot of unwanted eigenvalues appear, as shown in Fig:1,which shows that for large mesh size this shift is not goodchoice. Later different shifts were introduced, with real aswell as imaginary parts, and found to be effective. Thenumber of iterations taken by GMRES preconditioned CLSPM(1, π/4) gorws with linear rate with wave number. Thisfact is illustrated by spectrum of preconditioned Helmholtz,as shown in Fig:2, where eigen values are getting more closerto orign. Some near-origin eigenvalues affect the convergenceof solver. Deflation preconditioned, illustrated in next Sectin,is used to treat this drawback. A comparison of performanceof CSLP with different shifts in given in Table I, where shift(1, π/4) is the one which outperforms rest of choices of shiftsfor small as well large wave numbers.
TABLE ICOMPARISON OF GMRES NUMBER OF ITERATIONS BY CSLP WITH
Convergence of the Krylov subspace method is typicallyadversely affected by small eigenvalues, as seen in Fig: 2.The small eigenvalues need to be special treatment. Deflationis special type of preconditioner. Deflation is a techniquecommonly used to get rid of certain part of the spectrum,and to force the “unfair” eigenvalues not to participate inthe Krylov iterative method. In order to develop deflationpreconditioner, we consider the linear system
Ahuh = fh. (7)
For given a matrix Zh ∈ Cn×r, the deflation precondionersare the projections of type
Ph = Ih −AhQh, (8)
Proceedings of the World Congress on Engineering and Computer Science 2019 WCECS 2019, October 22-24, 2019, San Francisco, USA
of deflation vectors in matrix Zh forms an interest areaof research. Theoretically, eigenvectors gives ideal results,as they projects corresponding eigenvectors to zero. Sinceexact eigenvectors are impractical to compute, thereforemany problem-specific possibilities have been explored. Forthe problem under consideration, few alternatives have beenresearched in [25], [9] and [26]. Getting motivation fromproperty of resolving smaller error modes on coarser grids bymultigrid, we choose multigrid coarsegrid operator as defla-tion matrix Zh for our problem. This deflation preconditionercan be applied in combination with other preconditioners[27] and [28], and we have combined with CSLP as follows:
PhM(a, b)−1h Ahuh = PhM(a, b)−1
h fh. (9)
Next, we plot the spectrum of operator given in Eq: 9 in one-dimension as well as two-dimension in Fig: 3 and Fig: 4.These spectral plots show clustered behavior of eigenvaluesof deflated CSLP-preconditioned matrix in one dimensionand two dimension respectively.
Fig. 3. One Dimensional Spectrum with CSLP and Def.
Fig. 4. Spectrum with M(0, 1)
IV. NUMERICAL EXPERIMENTS
For all the experiments, u0 (zero vector) is used as initialguess. The mesh size h is chosen such that for a wave numberk, it satisfies relation kh ≤ 0.625 (equivalent to 10 grid
points per wave length). Iterations are stoppedn when theresidul meets the tolerance
‖rh‖ ≤ 10−5.
A. Results
The first numerical result, using deflation, is presentedin Table II, where CSLP is not used. This effort has beenmade to highlight affect of deflation preconditioner on itsown. The readings show a substantial reduction in numberof iterations and computational time. Subsequently, deflationis applied in combination with the CSLP, the first levelpreconditioner. A variety of shifts in CSLPj, in combinationwith defltaion, has been experimented and readings have beenrecorded and presented in Tables III, IV, V and VI for CSLP-shifts (0, 0) , (0, 1), (1, 1) and (1, π4 ) respectively. Suchcomparison is represented using consolidated bar plots givenin Fig: 6, where bar representing iterations taken by solverpreconditioned by CSLP and deflation is fairly smaller thanthe bar representing iterations taken by solver preconditionedby only CSLP. Comparison is performed with four differentchoices of shifts, comprehensible from figure. The two levelpreconditioned (CSLP and deflation perconditioner) solveris also tested for a very large wave number k = 200,and readings are presented in Table VII where inclusion ofdefltaion alongwith CSLP reduced the number of iterationssignificantly. Rate of reduction for shift (1, π4 ) is 5 times.Lastly, the velocity potential for wave number ranging fromk = 5 to k = 30 is plotted in Fig:5. Increasing wave numberclearly highlights the need of more grid-points for large wavenumber.
TABLE IINUMBER OF ITERATIONS BY GMRES AND DEFLATED GMRES
k N Dim. of A GMRES It. Time Def GMRES It. Time20 32 1089 74 00.79 13 00.1140 64 4225 200 08.63 15 00.4460 96 9409 405 51.95 17 01.1080 128 16641 607 202.30 20 02.44
100 160 25921 782 362.79 24 04.36
TABLE IIINUMBER OF ITERATIONS BY CSLP AND CSLP-DEFLATION WITH
M(0, 0)
k N Dim. A CSLPIt Time(S) CSLP-Def It Time(S)20 32 1089 20 00.25 07 00.1640 64 4225 71 02.86 10 00.7060 96 9409 154 17.28 13 02.3080 128 16641 257 66.02 16 05.26
100 160 25921 358 133.68 20 09.56
TABLE IVNUMBER OF ITERATIONS BY CSLP AND CSLP-DEFLATION WITH
M(0, 1)
k N Dim of A CSLPIt Time (S) Def CSLP It Time(S)20 32 1089 21 00.27 07 00.1640 64 4225 53 02.15 10 00.7560 96 9409 98 10.56 13 02.3280 128 16641 137 28.70 16 05.18
100 160 25921 175 56.93 20 09.73
Proceedings of the World Congress on Engineering and Computer Science 2019 WCECS 2019, October 22-24, 2019, San Francisco, USA
TABLE VNUMBER OF ITERATIONS BY CSLP AND CSLP-DEFLATION WITH
M(1, 1)
k N Dim A CSLP It Time Def. CSLP It Time20 32 1089 20 00.26 07 00.1640 64 4225 44 01.77 10 00.7360 96 9409 67 07.07 13 02.2880 128 16641 88 17.87 16 05.37
100 160 25921 108 33.44 21 10.16
TABLE VINUMBER OF ITERATIONS BY CSLP AND CSLP-DEFLATION WITH
M(1, π/4)
k N Dim A CSLP It Time Def. CSLP It Time20 32 1089 18 00.24 07 00.1640 64 4225 37 01.49 10 00.7460 96 9409 56 05.84 13 02.2780 128 16641 73 14.54 16 05.09
100 160 25921 88 26.88 20 09.64
TABLE VIINUMBER OF ITERATIONS BY CSLP AND CSLP-DEFLATION, DIFFERENT
SHIFTS M(a, b)
CSLP M(a, b) Dim A CSLP It t(s) D-CSLP It t(s)M(0, 0) 103041 948 1681 48 107M(0, 1) 103041 349 542 49 108M(1, 1) 103041 208 267 49 86M(1, π/4) 103041 169 248 49 83
V. CONCLUSION
In this paper, we discussed the ingredients of robust andefficient iterative solver for high wave number Helmholtzproblems. Need of preconditioner is highlighted and a criticalinvestigation of different preconditioners is presented. TheCSLP preconditioner is applied and found to be very effectiveto enhance the convergence of Krylov subspace methodsfor small wave number problem. Increasing wave numberstagnates convergence of CSLP preconditioned solver. Thedeflation is introduced and is used as a second-level incombination with CSLP, which not only pushes the smalleigenvalues to origin ( unwanted eigenvalues ), also helpsto achieve faster convergence fast. Specially when wavenumbers are large, the deflation method takes less iterationsas compared to the CSLP. It also reduces solve time for largewave number problem, which highlights the contribution ofthis paper.
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